[Paper Review] Comments on the large Nc behavior of light scalars

J. R. Peláez|ArXiv.org|Sep 26, 2005

Black Holes and Theoretical Physics8 references3 citations

TL;DR

This paper investigates the large-Nc behavior of light scalar resonances (σ, κ, f₀(980), a₀(980)) generated via unitarized one-loop Chiral Perturbation Theory. Unlike vector mesons (ρ, K*), which scale as conventional $ar{q}q$ states with width $ o O(1/N_c)$, light scalars exhibit vanishing meson-meson amplitudes and widths scaling as $O(N_c^{1/2})$ to $O(N_c)$, indicating a non-$ar{q}q$ dominant structure, likely dominated by two-meson or tetraquark components.

ABSTRACT

I review the large Nc behavior of light resonances generated from unitarized one-loop Chiral Perturbation Theory. In contrast with the rho or K*, the scalar behavior is at odds with a qqbar "dominant component". In fact, in the light scalar region, meson-meson amplitudes vanish as Nc increases. Also, the scalar widths, obtained from their associated poles, behave as O(Nc^{1/2})

Motivation & Objective

• To determine the large-Nc behavior of light scalar resonances generated from unitarized one-loop Chiral Perturbation Theory.
• To assess whether light scalars such as σ(600) and κ(800) can be interpreted as dominant $ar{q}q$ states in the large-Nc limit.
• To clarify the physical relevance of the $N_c \to \infty$ limit for understanding scalar meson structure.
• To investigate how the renormalization scale choice affects the large-Nc scaling of scalar poles and the interpretation of their dominant component.

Proposed method

• Uses the Inverse Amplitude Method (IAM) to unitarize one-loop ChPT amplitudes, generating resonances like σ, κ, and f₀(980) from meson-meson scattering amplitudes.
• Scales ChPT parameters ($L_i$, $f_0$, meson masses) according to large-Nc counting rules: $f_0 \sim \sqrt{N_c}$, $L_i \sim O(1)$ or $O(N_c)$.
• Analyzes pole positions ($\sqrt{s_{\text{pole}}} = M - i\Gamma/2$) in the second Riemann sheet to extract resonance masses and widths as functions of $N_c$.
• Compares the $N_c$-dependence of scalar resonances with that of well-established $ar{q}q$ states (ρ, K*) to test the $ar{q}q$ dominance hypothesis.
• Varies the renormalization scale $\mu$ between 0.5 and 1 GeV to assess uncertainty in the large-$N_c$ extrapolation.
• Examines the mathematical $N_c \to \infty$ limit and its physical relevance, noting that it may not reflect the true dominant component due to non-unique behavior and scale dependence.

Experimental results

Research questions

• RQ1 Do light scalar resonances such as σ(600) and κ(800) exhibit large-Nc scaling consistent with a dominant $ar{q}q$ component, like the ρ and K* mesons?
• RQ2 How does the width of light scalar resonances scale with $N_c$ in the unitarized ChPT framework?
• RQ3 Why does the $N_c \to \infty$ limit fail to reliably determine the dominant component of light scalars despite being mathematically well-defined?
• RQ4 Can the observed $N_c$-dependence of scalar amplitudes and poles be explained by a two-meson or tetraquark component instead of $ar{q}q$?
• RQ5 How sensitive is the large-Nc behavior of scalar poles to the choice of renormalization scale $\mu$, and what does this imply for physical interpretation?

Key findings

• The meson-meson scattering amplitudes for light scalars vanish as $N_c$ increases, indicating a lack of $ar{q}q$ dominance.
• The widths of light scalar resonances scale as $O(N_c^{1/2})$ to $O(N_c)$, in stark contrast to $ar{q}q$ states whose widths scale as $O(1/N_c)$.
• The $\rho$ and $K^*$ vector resonances behave as expected for $\bar{q}q$ states: mass remains $O(1)$ and width $O(1/N_c)$, confirming the method's consistency.
• The $N_c \to \infty$ limit is not unique and can yield unphysical results (e.g., negative mass squares or poles at infinity), limiting its physical relevance.
• The renormalization scale $\mu$ choice between 0.5 and 1 GeV significantly affects the large-$N_c$ behavior of scalar poles, implying that robust conclusions must be drawn near $N_c = 3$, not at infinity.
• The dominant component of light scalars is not a $ar{q}q$ state; instead, two-meson or tetraquark components are qualitatively favored, as they dissolve into the continuum at large $N_c$.